On Chaotic Color Image Encryption Using an Improved Hill Cipher with a Non-Square Pseudo-Random Key Matrix

The three most widely used chaotic maps in the field of cryptography are utilized in this novel approach, which maintains strong resistance against brute force attacks due to their large key size and high sensitivity to initial parameters. Additionally, this novel cryptosystem accepts six pixels as input via a non-square encryption lattice and returns three Pseudo-Random pixels for the encrypted image reconstruction. The axes listed below serve as instruments for explaining this novel algorithm:

2.1 Chaotic sequences selection

Because of their extraordinary sensitivity to initial conditions and easy configuration, our algorithm suggested using the three most researched maps in cryptographic schemes [16, 17]: the logistic, skew tent, and A.J. maps, as illustrated in Figure 1.

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Figure 1. Chaotic map selected

2.1.1 The logistics maps

The map u_n [4], defined as a sensitive recursive formula of a simple second-degree polynomial by Eq. (1), generates real sequences. The bifurcation diagram illustrating the chaotic aspect of the logistics map is given in Figure 2.

$\left\{\begin{array}{l}u_0 \in[0.5 ; 1], \mu \in[3.75 ; 4] \\ u_{n+1}=\mu u_n\left(1-u_n\right)\end{array}\right.$                (1)

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Figure 2. Bifurcation diagram of the logistics map

Values for the quantitative assessment of chaotic performance are represented by the Lyapunov exponent. Stated otherwise, the presence of a positive value for the Lyapunov exponent, depicted in Figure 3, indicates that the nonlinear map is sensitive to initial conditions, demonstrates improved chaotic behaviors with respect to the unpredictability characteristic, and also represents the best chaotic performances. From a mathematical perspective, the Lyapunov exponent Eq. (2), looks like this:

$\lambda=\frac{1}{n} \sum_{i=0}^n \operatorname{Ln}\left(\left|f^{\prime}\left(u_i\right)\right|\right)$           (2)

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Figure 3. Lyapunov of the logistic map

2.1.2 The skew tent map

The skew tent map (SKTM) (V_n) [18-20] will be redefined as the Eq. (3).

$\left\{\begin{array}{c}\mathrm{v}_0 \in[0 ; 1], \mathrm{p} \in[0.5 ; 1] \\ \mathrm{v}_{\mathrm{n}+1}=\left\{\begin{array}{c}(\mathrm{p})^{-1} \mathrm{v}_{\mathrm{n}}, \text { if } 0<\mathrm{v}_{\mathrm{n}}<\mathrm{p} \\ (1-\mathrm{p})^{-1} \left(1-\mathrm{v}_{\mathrm{n}}\right), \text { if } \mathrm{p}<\mathrm{v}_{\mathrm{n}}<1\end{array}\right.\end{array}\right.$        (3)

This card possesses a highly significant chaotic component for a color image encryption system, as seen by the bifurcation diagram in Figure 4.

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Figure 4. Bifurcation diagram of the SKTM

2.1.3 The A.J. map

The A.J. map formula of this map [21, 22] given in Eq. (4), has a great sensitivity to beginning circumstances, chaotic behavior, and ease of implementation are the main reasons it was chosen; its Lyapunov exponent is on the order of $(p * \log (2)>\log (2)$ with $p>1)$.

$\left\{\begin{array}{c}\mathrm{w}_0 \in[1 /(1+p) ; p /(1+p)], \mathrm{p} \in[1.65 ; \varphi] \\ \mathrm{f}\left(\mathrm{w}_{\mathrm{n}}\right)=\mathrm{w}_{\mathrm{n}+1}\left\{\begin{array}{c}\mathrm{p}^2 \mathrm{w}_{\mathrm{n}}, \text { if } 0 \leq \mathrm{w}_{\mathrm{n}} \leq(1+\mathrm{p})^{-1} \\ \mathrm{p}-p \mathrm{w}_{\mathrm{n}}, \text { if } (1+\mathrm{p})^{-1} \leq \mathrm{w}_{\mathrm{n}} \leq 1\end{array}\right.\end{array}\right.$                     (4)

The integration of three chaotic maps in a hybrid approach enables the derivation of all critical parameters required to optimize the efficiency and effectiveness of our innovative architecture. In Figure 5, the chaotic trajectory for an initial value of 0.75 is shown.

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(a) Trajectory of the initial value 0.75

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(b) Initial conditions sensitivity

Figure 5. Trajectory of the initial value 0.75 and initial conditions sensitivity

These chaotic maps were chosen for our cryptosystem for the following reasons:

• Sensitivity to initial conditions.

• Random appearance.

• Ease of implementation.

2.1.4 Compare the three chaotic maps

Despite the reliability of the three chaotic maps, there are minimal differences between them, as summarized in Table 1. We quote:

Table 1. Mapping function comparison

Chaotic Card Advantages.

• Low Computational Cost: Ideal for embedded devices. 

• Large Key Space: Depends on initial condition and control parameters, ensuring high security. 

• Adaptability: Can be integrated with other systems, such as hybrid chaos or quantum encryption.

Limitations.

• Vulnerability to statistical analysis attacks: If poorly implemented, it may require post-processing to enhance security.

• Risk of short cycles: Improper selection of control parameters can lead to short periodic sequences.

The combination of these three chaotic maps will be used to generate all the parameters necessary for the proper functioning and operation of our new technology.

2.2 Building encryption tools

This first phase acts at the pixel level by implementing an improved Vigenère technique that uses two large S-Boxes. For the smooth running of this part, several parameters will be developed.

• (WC) of size (3 nm;5) confusion/diffusion table.

• (WB) of size (3 nm;5) confusion/diffusion binary table.

• (WS1); (WS2) two substitution tables of size (t 256), with (t) an integer calculated from the Pseudo-Random vectors and under the control of the binary decision vector.

The construction of all these parameters is illustrated in Figure 6.

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Figure 6. Encryption tools generation

2.2.1 (WC) Confusion/diffusion table design

The (WC) table of size (3 nm;5) with coefficients in (G₂₅₆)  is used to facilitate communication and diffusion among the original image pixels. It has been spoken regarded as an encryption subkey. The construction of this table is described by Algorithm 1.

All the table (WC) columns are independent pseudorandom vectors from each other.

2.2.2 (WB) Binary control tables construction

The (WB) binary table of size (24 nm;2) employed to control any cipher operation used in our new system, is developed by Algorithm 2.

Each column in this table represents the binary vector that will serve as the control mechanism for one or more cipher processes.

2.3 Substitution table development

The creation of large S-Boxes for modifying pixel values in our technique is based on generating many Pseudo-Random permutations controlled by decision tables. This involves multiple linear congruence generators (LCGs) with maximum periods. A mathematical reminder on LCGs is added:

2.3.1 Mathematics reminder

In computer science, particularly in cryptography, LCGs are a highly esteemed type of pseudorandom number generator that are frequently employed. Their effectiveness and simplicity are respected. An LCG occupies a simple linear recurrence relation to generate an assortment of pseudorandom integers. Algorithm 3 contains the general algorithm for an LCG.

(S₀; a; b; m) are referred to as generator parameters. A well-designed LCG features a maximal period of (m) Additionally, the randomness of the generated numbers is assured, with Hull & Dobell's theorem offering the essential and comprehensive condition for an LCG to exhibit Pseudo-Random behavior.

In computer science, LCGs serve as essential tools for generating pseudorandom numbers. They are a popular option due to their efficiency and simplicity, but special precautions must be taken to guarantee the security and quality of the created sequences. Current developments in this area attempt to overcome their shortcomings and exceed the changing needs present in modern applications.

2.3.2 Hull-Dobell theorem 1962

All LCGs that meet the parameters outlined in the previous theorem will have a maximal period of mm, as long as the conditions are fulfilled by the connection presented in Algorithm 4.

2.3.3 Particular case (m = 2^k)

Consider the particular case of LCG, with m = 2^k, depicted in Algorithm 5.

This generator has a period of 255 if and only if the theorem in Algorithm 6 is verified. A detailed example that depicts the distribution of the LCG in G₁₆ is given in Table 2.

Table 2. Example of LCG distribution in (G₁₆) 

We highlight where there is Pseudo-Randomness in the arrangement of LCG. These generators are going to be used for manufacturing two replacement tables, with dimensions of $(\mathrm{t}>256,256)$, labeled (WC1) and (WS2). As each generator has a parameter $\left(\mathrm{s}_0 ; \mathrm{a} ; \mathrm{b} ; 256\right)$ fulfilling Hull Dobell formula expressed by Eq. (6) will be stored in a table (WL) of size $(6 ; t)$.

2.3.4 Affine function

Any affine application is defined by Eq. (5).

$\begin{gathered}\text { 1. } \mathrm{f}:\left(\mathrm{G}_{256}\right) \rightarrow\left(\mathrm{G}_{256}\right) \\ \text { 2. } \mathrm{x}: \mapsto \bmod (\mathrm{c} * \mathrm{x}+\mathrm{d} ; 256)\end{gathered}$    (5)

This function is a bijection if and only if:

(1) c: is an odd integer, as illustrated in Figure 7;

(2) d: is an arbitrary integer, as illustrated in Figure 8.

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Figure 7. Distribution of the LCG

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Figure 8. Influence of the affine function

2.3.5 S-Box size calculation (tb)

The parameter (tb), that reflects the size of each S-Box is established by the algorithm beneath Algorithm 7. We notice that: 260 ≤ t < 360.

The size of the two S-Boxes relies on the size of the image, the pseudorandom vectors, and the control binary vector WB(: 2).

2.3.6 (WL) Parametric table construction

The (LCG) referred to the recurrent relation depicted in Algorithm 8 meets the Hull & Dobell theorem's criteria.

The three parameters $\left(\mathrm{s}_0, \mathrm{a}, \mathrm{b}\right)$ of an LCG used in our algorithm for the (WS1) substitution tables development of dimension ($t \geq 256 ; 256$), and ($c ; d$) for affine function, is stored in the $(5 ; \mathrm{t})$ array $(\mathrm{WL})$ following the key points:

• The starting generator values in (G₂₅₆), referred as $\left(\mathrm{s}_0\right)$, are inserted as a pseudorandom seed in the first line.
• The multiplier parameter values $(a=(4 t+1) \% 256)$ are inserted in the second line.
• The bias parameter values $(\mathrm{b}=(2 \mathrm{~h}+1) \% 256)$ are inserted in the third line.
• The multiplier of the bijective affine function values $(c=(2 q+1) \% 256)$ are inserted in the fourth line.
• The duplicator of the bijective affine function values (d) is inserted in the fifth line.

Algorithm 9 describes how to develop the table (WL).

Table 3. Pseudo-Random table (WL) example

The aforementioned layout is exceedingly sensitive to any alteration in any private key component, because is managed by Pseudo-Random vectors $\operatorname{WB}(: 2)$; this increases the system's perseverance.

An example in $\left(\mathrm{G}_{16}\right.$ and $\left.\mathrm{t}=16\right)$, of Pseudo-Random table $\mathrm{(WL)}$ is given by Table 3.

An additional set $L C G$ becomes an example in Algorithm 10.

The values generated by this LCG are given in Table 4.

Table 4. Additional LCG example calculated values

Nonlinearity of expanded permutations. In cryptanalysis, a differential attack examines the propagation of input differences through a cipher's output. This technique is especially potent against block ciphers, which rely on S-Boxes and P-boxes to achieve confusion and diffusion.

Each permutation (Pi) is nonlinear, which ensures that the S-Box (BW1) is a substitution table and is also nonlinear.

We prove that (P1) is nonlinear: if (P1) is linear, then we will have: $\forall x, y P 1(x+y)=P 1(x)+P 1(y)$.

So, If: $\exists x, y P 1(x+y) \neq P 1(x)+P 1(y)$.

Then (P1) is nonlinear, we consider the table (LN) of size ($3 ; 256$) with a rank offset (k) defined as follows:

• First line contains $\mathrm{x}+\mathrm{y}$.
• Second line contains P1 $(\mathrm{x}+\mathrm{y})$.
• Third line contains P1 $(x) \oplus \mathrm{P} 1(y)$.

The construction of the table (LN) is given by Algorithm 11.

In our example, p = 99%. So, our permutation is 99% different from any linear application. The computation steps are shown in Tables 5-7.

Table 5. Compute f(x) 

Table 6. Compute f(x) + f(y)

Table 7. Compute f(x + y)

Therefore, the permutation used in the example is nonlinear. In our example, $\mathrm{p}=90,92 \%$. So, our (P1) permutation is $91 \%$ different from any linear application.

2.3.7 (WS1) Novel S-Box development

The creation of the S-Box (WS1) is fully governed by the table (TC1) of size (t, 6). All columns of the table (TC1) are permutations in $\left(\mathrm{G}_{256}\right)$.

(TC1) Construction. Table (TC1) enables the construction of the first $\mathrm{S}-$ Box under Pseudo-Random conditions by utilizing LCGs to create a permutation. The permutations are described below:

• The first column is given by the ($\operatorname{Pr} 1$) permutation derived by the (WC(: 2)) first (t) ascendants reordered values.
• The second column is given by the ($\operatorname{Pr} 2$) permutation derived by the (WC(: 3 )) first (t) ascendants reordered values.
• The third column is given by the ($\operatorname{Pr} 3$) permutation derived by the (WC(: 5)) first (t) ascendants reordered values.
• The fourth column is given by the ($\operatorname{Pr} 4$) permutation derived by the (WC(: 1)) first (t) ascendants reordered values.
• The fifth column is given by the ($\operatorname{Pr} 5$) permutation derived by the (WC(: 4)) first (t) ascendants reordered values.
• The sixth column is given by the ($\operatorname{Pr} 6$) permutation derived by the (WB(: 1)) first (t) ascendants reordered values.

An example of permutation construction is given in Table 8.

Table 8. Permutation construction example 12

Construction of the table (TC1) of size (t;6) is given by Algorithm 12.

2.3.8 (TC1) Description

The table (TC1), which specifies the parameters of the LCG to be selected from the table (WL), will be used to generate the first S-Box (WS1) with a size of (t > 256; 256). This SBox will function as a substitution table. The dimensions (t; 6) of this table (TC1) are specified by:

LCG construction.

1. The initial condition $\left(s_0\right)$ values position from the table (WL) are inserted in the first column of the table (TC1).

2. The parameter (a) values position from the table (WL) are inserted in the second column of the table (TC1).

3. The parameter (b) values position from the table (WL) are inserted in the third column of the table (TC1).

Affine function parameter.

4. The parameter (c) values position from the table (WL) are inserted in the fourth column of the table (TC1).

5. The parameter (d) values position from the table (WL) are inserted in the fifth column of the table (TC1).

Ordering column of the obtained permutation.

6. The numbers of the rows where to place the obtained generator are inserted in the sixth column of the table (TC1).

An example about how to construct the table (TC1) in $\left(\mathrm{G}_8\right)$ with t = 10 is given in Table 9.

Table 9. Example of (TC2) in $\left(\mathrm{G}_8\right)$ and t = 12

2.3.9  (WS1) Construction

The table's LCG's (WL) parameters are chosen in the first three columns. The values produced can be changed by applying the coefficients of the affine function to the generator in the fourth and fifth columns. The values of the created permutation should be positioned on the line indicated by the seventh column. Consequently, the procedure below operates for building the first S-Box (SW1) based on Algorithm 13.

2.3.10 (WS2) New S-Box development

The construction of the S-Boxe (WS2) is fully governed by the (t, 6) array (TC2). All columns of the array (TC2) are permutations in $\left(\mathrm{G}_{256}\right)$.

(TC2) Design. The columns of this table are created following these operations:

• The first column is given by the (Qr1) permutation which results from (t) sorted first values of the list (WB(:1)) in ascending order.
• The second column is given by the (Qr2) permutation which results from the (t) sorted first values of the list (WC(:5)) in ascending order.
• The third column is given by the (Qr3) permutation which results from (t) sorted first values of the list (WC(:2)) in ascending order.
• Then, the fourth column is given by the (Qr4) permutation which results from the (t) sorted first values of the list (WC(:3)) in ascending order.

The construction of the array (TC2) of size ($t \geq 256; 6$) is given by the Algorithm 14. An example about how to construct the table (TC2) in $\left(\mathrm{G}_8\right)$ for t = 10 is given in Table 10.

Table 10. (WL) table construction example in (G₁₆)

(TC2) Description.

The construction of the second S-Box (WS2), of size $(t \geq 256,256)$ and which will serve as a replacement table, is defined by the following procedures:

LCG construction.

1. The initial condition ($\mathrm{s}_0$) values from the table (WL) are inserted in the first (TC2).

2. The parameter (a) values from the table (WL) are inserted in the second (TC2).

3. The parameter (b) values position from the table (WL) are inserted in the third column (TC2).

Affine function parameter.

4. The parameter (c) values position from the table (WL) are inserted in the fourth column (TC2).

5. The parameter (d) values position from the table (WL) are inserted in the fifth column (TC2).

Ordering column of the obtained permutation.

6. The numbers of the rows where to place the obtained generator are inserted in the sixth column (TC2).

2.3.11 (WS2) Design

This novel (WS2) is provided by Algorithm 15.

Table 11 depicts the used table example in (G₁₆) for the purpose of constructing the (WS1) S-Box, and Figure 9 depicts an example of the table (TC1).

Table 11. (WS1) S-Box construction example in G₁₆

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 Figure 9. (TC1) construction example

2.3.12 New pixel transformation using both S-Boxes

Let's transform the pixel X(i) of rank (i) into Y(i) using the (WG) new confusion formula, based on two S-Boxes [23, 24] and the pseudorandom table (WC) under the (WB(: 2) control vector is given by the Algorithm 16.

The two substitution tables are used together with the binary decision vector (WB(:2)) to install the obfuscation and diffusion process. Thus, a simple change of one initial value of the chaotic sequence results in a new control vector, which in turn results in a new obfuscation equation and thus another scrambled image.

2.4 Plain image preparation

Most importantly, each image must be prepared to facilitate the encryption process. The following steps explain this preparation phase.

2.4.1 Switching to image vector notation

Coupling of the three RGB by application of the XOR operation with the confusion vector (WC) into the vector $X\left(x_1, \ldots \ldots ., x_{3 n m}\right)$ of size $(1,3~n m)$, need to extract and convert them into three vectors $(\mathrm{Vr}),(\mathrm{Vg})$, and $(\mathrm{Vb})$ of dimensions (1 nm) each. The operation is illustrated by Algorithm 17.

The decision vector $(WB(: 2))$ controls the switch to vector notation. By taking this step, the high correlation between pixels is successfully reduced, creating a strong encryption that can withstand statistical and brute force attacks. The diagram in the subsequent Figure 10 displays this early phase.

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Figure 10. First round image encryption diagram

Pseudo-Random vectors and the two S-Boxes enable the vector optimization phase of the original image to generate a partially encrypted image, whose statistical properties ensure its resilience against statistical and frequency-based attacks. However, to address differential attacks, we will enhance security by employing diffusion functions through the implementation of the new Hill cipher technique using a non-square matrix.

2.4.2 The initialization value computation

The initialization value calculation is necessary to alter the startup pixel for the following iterations and to ensure the encryption process starts correctly. This initialization value is determined by Algorithm 18.

It is essential to keep in mind that the control vector (WB) and the initial image share an intimate connection with the computed initialization value. A different encrypted image will develop from any alterations made to the private key parameters or source image, which will also alter the initialization values.

2.4.3 Encryption process

In order to prevent any differential attack on the system, the initial round is handled [25-27]. This is a relatively short step that uses the S-Box extensively to accomplish diffusion and confusion. This phase is depicted by the diagram below:

• Subdivision of the vector $(X)$ into $(\mathrm{nm})$ sub-blocks each containing three pixels.
• Subdivision of the vector $(W C(: 3))$ into $(\mathrm{nm})$ sub-blocks each containing three pixels.
• Application of the non-square encryption matrix on the six-pixel block.
• Obtaining three output pixels.
• Applying an external broadcast again to connect the next clear block to the already encrypted block.
• Repeat the process until the last block.

2.4.4 A review of the new technique's components

Steps 1 and 2: This first phase, depicted in Figures 11 and 12, consists of subdividing the original image and some Pseudo-Random vectors into sub-blocks of three pixels.

Steps 3, 4 and 5: The seed pixel is modified using an Initialization Value. Confusion/Diffusion: Perform the first internal confusion/diffusion operation on the block. (WC) Augmentation: The block is augmented by adding three Pseudo-Random values obtained from the table.

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Figure 11. Subdividing the original image into sub-blocks

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Figure 12. Pseudo-Random vectors to sub-blocks

Steps 6 and 7: These two steps are seen in Figure 13.

The permutation within the block is a displacement of rank $P D(i)$ or $P F(i)$ depending on the value of the binary control vector $B(: 1)$. The mathematical expressions of used parameters' permutation within a block is given in Table 12.

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Figure 13. Sub-blocks permutation application

Table 12. Mathematical expressions of used parameters’ permutation within a block

Step 8: External diffusion.

With the support of the Pseudo-Random matrix (M) having dimension (3;3), whose is explained by the relation below, the diffusion process between the next clear block and the current cipher block is installed in this last encryption stage of development [18, 19]. Let (M) be the Pseudo-Random diffusion matrix, which is given in Figure 14.

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Figure 14. Pseudo-Random diffusion matrix

2.5 Encrypted image decryption

We have developed a symmetric encryption technique. Consequently, throughout the decryption process [28-30], all encryption parameters will be employed utilizing reciprocal functions, beginning with the final encrypted block. The decryption procedure adheres to this timetable.

2.5.1 Reverse of the Hill method

$\left(\begin{array}{l}Z 1 \\ Z 2 \\ Z 3\end{array}\right)=H^t\left(\begin{array}{l}P 1 \\ P 2 \\ P 3 \\ P 4 \\ P 5 \\ P 6\end{array}\right) H\left(\begin{array}{l}Z 1 \\ Z 2 \\ Z 3\end{array}\right)=H H^t\left(\begin{array}{l}P 1 \\ P 2 \\ P 3 \\ P 4 \\ P 5 \\ P 6\end{array}\right)$

$H^t:$ a positive definite symmetric matrix, therefore invertible.

In effect:

$\begin{gathered}X H H^t X^t=(X H) *(X H)^t>0 \\ H\left(\begin{array}{l}Z 1 \\ Z 2 \\ Z 3\end{array}\right)=\mathrm{HH}^{\mathrm{t}}\left(\begin{array}{l}\mathrm{P} 1 \\ \mathrm{P} 2 \\ \mathrm{P} 3 \\ \mathrm{P} 4 \\ \mathrm{P} 5 \\ \mathrm{P} 6\end{array}\right) \\ \left(\begin{array}{l}\mathrm{P} 1 \\ \mathrm{P} 2 \\ \mathrm{P} 3 \\ \mathrm{P} 4 \\ \mathrm{P} 5 \\ \mathrm{P} 6\end{array}\right)=\left(\mathrm{HH}^{\mathrm{t}}\right)^{-1} * \mathrm{H} *\left(\begin{array}{l}\mathrm{Z} 1 \\ \mathrm{Z} 2 \\ \mathrm{Z} 3\end{array}\right)\end{gathered}$

Put $\mathrm{K}=\left(\mathrm{HH}^{\mathrm{t}}\right)^{-1} * \mathrm{H} \quad\left(\begin{array}{l}\mathrm{P} 1 \\ \mathrm{P} 2 \\ \mathrm{P} 3 \\ \mathrm{P} 4 \\ \mathrm{P} 5 \\ \mathrm{P} 6\end{array}\right)=\mathrm{K} *\left(\begin{array}{l}\mathrm{Z} 1 \\ \mathrm{Z} 2 \\ \mathrm{Z} 3\end{array}\right)$

2.5.2 S-Box decryption matrix

Generally speaking, any S-Box used in a cipher system can be reversed by Algorithm 19 to be used in the decryption process [20, 31]. An example is depicted in Table 13.

Table 13. S-Box decryption matrix example

We can obtain (x), by applying the same logic of Vigenere’s conventional technique using Algorithm 20.

This formula allows us to recalculate the reciprocal function of the confusion function using the two S-Boxes.

• Confusion reciprocal function

This novel encryption employs an involutive diffusion function, as depicted in Algorithm 21. It is consequently comparable to its reciprocal.

• Diffusion Reciprocal function

The diffusion function is, by construction, an involutive function; consequently, it is identical to its reciprocal.

The chaotic maps produced by our research exhibit an extraordinary sensitivity to initial conditions. Furthermore, because our private key is so big, the system is protected from brute force attacks [18]. The randomness introduced by the permutation further adds to the potential attack’s complexity. At last, strong chaining mechanisms installed in both towers provide protection against known attacks against the new encryption method.

4.1 The benefits of the process

Our approach is suitable for images of all sizes and formats. Non-commutative algebraic operations enhance the system's robustness. The Pseudo-Random size of the substitution table makes it more difficult to reconstruct the employed S-Boxes. The structure of the S-Box, combined with its operation governed by a chaotic decision vector, adds further complexity to attacking our method. Moreover, encryption keys generated from chaotic maps are extremely sensitive to initial conditions, making it hard to retrieve the actual key employed.

4.2 The approach restrictions

The restrictions of our methodology are mostly dictated by the bounds of the S-Box building and chaotic map selection processes, in addition to the Pseudo-Random size of the S-Boxes that are formed.